In summary, the gradient of a function is a vector that gives the rate of change of the function in a given direction. It is perpendicular to the level curves of the function and the direction of steepest ascent. This can be visualized by imagining walking along a level path on a hill, where the steepest slope is perpendicular to the path. The concept of steepest descent and conjugate gradient methods can be further understood through resources like the article from Better Explained and videos from Khan Academy.
Can this(image) be used as a proof that the direction of gradient gives the direction of steepest ascent(in 2D).Am I understanding it right ?.The function 'f' in the image is a scalar valued function.Please explain.
the gradient is a vector whose dot product with a given direction vector gives the rate of change of the function in that direction. hence it dots to zero along a direction where the function is constant, and hence the gradient is perpendicular to the level curves (for a function defined on the plane) of the function. It seems obvious that the direction in which the function increases fastest is perpendicular to the level curve through a given point. just imagine you are walking along a level path cut into the side of a hill, wouldn't the slope of the hill be greatest in a direction perpendicular to the path?
